Solution of an optimal control problem in a distributed-parameter system

This paper treats the problem of optimal control of a typical distributed-parameter system governed by a heat conduction equation. The problem is to minimize, the deviation of the temperature distribution from the assigned distribution at a given time. Two methods are shown for the solution of the optimal control problem. One is the variational method, and the other consists of reducing the problem to a linear or nonlinear programming problem. Upon use of the variational technique, Fredholm's integral equation of the first kind is derived as a necessary condition for the optimal control. By replacing the minimization of a functional by the minimization of a function of many variables, numerical solutions can be obtained by using the technique of linear or nonlinear programming.